A Topological Approach to Left Eigenvalues of Quaternionic Matrices
M.J. Pereira-S´aez† (joint work with E. Mac´ıas-Virg´os∗)
†Departamento de Econom´ıaAplicada II, Universidade da Coru˜na
∗Departamento de Xeometr´ıae Topolox´ıa, Universidade de Santiago de Compostela
Sevilla, September 9-14, 2012 1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 2 / 69 Left eigenvalues
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 3 / 69 Left eigenvalues Notion
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 4 / 69 Left eigenvalues Notion
Notion Definition
Let A ∈ Mn×n(H), a quaternion λ ∈ H is said to be a left eigenvalue of the matrix A if Av = λv
for some vector v ∈ Hn, v 6= 0. Equivalently, A − λI is a singular matrix.
For a matrix A ∈ Mn×n(H), we denote σl (A) its left spectrum, i.e. the set of left eigenvalues.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 5 / 69 Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2.
Left eigenvalues Notion
About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2.
Left eigenvalues Notion
About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2.
Left eigenvalues Notion
About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009].
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 Left eigenvalues Notion
About the existence of left eigenvalues: Wood [Bull. Lond. Math. Soc., 1985] proved that any n × n quaternionic matrix A has at least one left eigenvalue. Huang and So [Linear Algebra Appl., 2001] proved that a 2 × 2 matrix may have one, two or infinite left eigenvalues. A different proof was presented by EMV and MJPS [Electron. J. Linear Algebra, 2009]. Both proofs are algebraic in nature and seemingly difficult to generalize for n > 2.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 6 / 69 * λ ∈ H is on σl (A) iff the matrix A − λI is not invertible.
Left eigenvalues Notion
– There is no systematic study of left eigenvalues for n > 2. – In this talk we try a topological approach that could be generalized to other dimensions.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 7 / 69 Left eigenvalues Notion
– There is no systematic study of left eigenvalues for n > 2. – In this talk we try a topological approach that could be generalized to other dimensions.
* λ ∈ H is on σl (A) iff the matrix A − λI is not invertible.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 7 / 69 Left eigenvalues Study’s determinant
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 8 / 69 Left eigenvalues Study’s determinant
Study’s determinant It is possible to generalize to the quaternions the norm | det | (that is, with real values) of the complex determinant.
Definition
Let the quaternionic matrix A ∈ Mn×n(H) be decomposed as A = X + jY with X , Y ∈ Mn×n(C). We shall call Study’s determinant of A the non-negative real number Sdet(A) := (det c(A))1/2, X −Y where c(A) is the complex matrix c(A) = ∈ M ( ). Y X 2n×2n C
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 9 / 69 Left eigenvalues Study’s determinant
Proposition Study’s determinant Sdet is the only functional that verifies the following properties : 1 Sdet(AB) = Sdet(A) · Sdet(B); 2 if A is a complex matrix then Sdet(A) = | det(A)|.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 10 / 69 Left eigenvalues Study’s determinant
The following immediate consequences are very useful for computations. Corollary
1 Sdet(A) > 0 if and only if the matrix A is invertible; −1 2 let A and B = PAP be similar matrices, then Sdet(A) = Sdet(B); 3 Sdet(A) does not change when a (right) multiple of one column is added to another column. 4 Sdet(A) does not change when a (left) multiple of one row is added to another row; 5 Sdet(A) does not change when two columns (or two rows) are permuted.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 11 / 69 Left eigenvalues Study’s determinant
We shall need the following property too Proposition
For any matrix formed by two boxes A, B of order m and n respectively, it holds that A 0 Sdet = Sdet(A) · Sdet(B). ∗ B
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 12 / 69 Left eigenvalues Characteristic map
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 13 / 69 Proposition
The spectrum σl (A) is compact.
Left eigenvalues Characteristic map
Characteristic map
Definition
The map µ : H → H is a characteristic map of the matrix A ∈ Mn×n(H) if, up to a constant, its norm verifies
|µ(λ)| = Sdet(A − λI)
for all λ ∈ H.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 14 / 69 Left eigenvalues Characteristic map
Characteristic map
Definition
The map µ : H → H is a characteristic map of the matrix A ∈ Mn×n(H) if, up to a constant, its norm verifies
|µ(λ)| = Sdet(A − λI)
for all λ ∈ H.
Proposition
The spectrum σl (A) is compact.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 14 / 69 Left eigenvalues Characteristic map
For example, let D = diag(q1,..., qn) be a diagonal matrix. Then
µ(λ) = (q1 − λ) ··· (qn − λ)
is a characteristic map for D.
For a triangular matrix it is the same.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 15 / 69 Spectrum of 2 × 2 matrices
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 16 / 69 Spectrum of 2 × 2 matrices
Remark The left spectrum is not invariant by similarity. However, if P is an invertible real matrix then Sdet(A − λI) = Sdet(PAP−1 − λI), hence A and PAP−1 have the same characteristic maps.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 17 / 69 if bc = 0 (triangular matrix), σl (A) = {a, d};
µ(λ) = (d − λ)(a − λ).
we study now the bc 6= 0 case.
Proposition
Computing σl (A) is equivalent to finding the roots of a characteristic map like
µ(λ) = c − (d − λ)b−1(a − λ).
Spectrum of 2 × 2 matrices
Let a b A = ∈ M ( ), c d 2×2 H Then,
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 18 / 69 we study now the bc 6= 0 case.
Proposition
Computing σl (A) is equivalent to finding the roots of a characteristic map like
µ(λ) = c − (d − λ)b−1(a − λ).
Spectrum of 2 × 2 matrices
Let a b A = ∈ M ( ), c d 2×2 H Then,
if bc = 0 (triangular matrix), σl (A) = {a, d};
µ(λ) = (d − λ)(a − λ).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 18 / 69 Spectrum of 2 × 2 matrices
Let a b A = ∈ M ( ), c d 2×2 H Then,
if bc = 0 (triangular matrix), σl (A) = {a, d};
µ(λ) = (d − λ)(a − λ).
we study now the bc 6= 0 case.
Proposition
Computing σl (A) is equivalent to finding the roots of a characteristic map like
µ(λ) = c − (d − λ)b−1(a − λ).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 18 / 69 Spectrum of 2 × 2 matrices
Proof .– The rank of a matrix is invariant if the matrix is replaced by another one obtained by adding to a column a right multiple of the other column.
a − λ b 0 b A − λI = ∼ . c d − λ c − (d − λ)b−1(a − λ) d − λ
The problem of finding the roots of a quaternionic polynomial like µ(λ) was completely solved by Huang and So [Linear Algebra Appl., 2001] by purely algebraic methods.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 19 / 69 Spectrum of 2 × 2 matrices
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 20 / 69 Spectrum of 2 × 2 matrices Topological degree
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 21 / 69 Spectrum of 2 × 2 matrices Topological degree
Topological degree
We want to apply the following well known global result Theorem Let M be a connected closed oriented manifold, let µ: M → M be a differentiable map of degree k. Let m ∈ M be a regular value such that the differential µ∗λ preserves the orientation for any λ in the fiber µ−1(m). Then µ−1(m) is a finite set with k points.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 22 / 69 Spectrum of 2 × 2 matrices Linearization
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 23 / 69 Spectrum of 2 × 2 matrices Linearization
Linearization
Now, in order to obtain the regular values of µ, we show how can we compute the differential µ∗. The existence of the multiplicative norm |q| = (qq¯)1/2 on H guarantees that the usual proof of the Leibniz rule still works.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 24 / 69 Let f : H → H be a differentiable map such that f (λ) 6= 0 for all λ ∈ H. Let (1/f ): H\{0} → H be the map given by (1/f )(λ) = f (λ)−1. Then the differential is given by
−1 −1 (1/f )∗λ(X ) = −f (λ) f∗λ(X )f (λ) .
Spectrum of 2 × 2 matrices Linearization
Let f , g : H → H be two differentiable maps. Then the differential of the product is given by
(fg)∗λ(X ) = f∗λ(X )g(λ) + f (λ)g∗λ(X ).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 25 / 69 Spectrum of 2 × 2 matrices Linearization
Let f , g : H → H be two differentiable maps. Then the differential of the product is given by
(fg)∗λ(X ) = f∗λ(X )g(λ) + f (λ)g∗λ(X ).
Let f : H → H be a differentiable map such that f (λ) 6= 0 for all λ ∈ H. Let (1/f ): H\{0} → H be the map given by (1/f )(λ) = f (λ)−1. Then the differential is given by
−1 −1 (1/f )∗λ(X ) = −f (λ) f∗λ(X )f (λ) .
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 25 / 69 Proposition
A polynomial map like µ and the power map λ2 are homotopic, hence they have the same topological degree, which equals 2.
Spectrum of 2 × 2 matrices Linearization
We remark that the characteristic map µ: H → H, µ(λ) = c − (d − λ)b−1(a − λ) can be extended to a continuous map
µ: S 4 → S 4
on the sphere S 4 = H ∪ {∞}, because lim |µ(λ)| = ∞. |λ|→∞
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 26 / 69 Spectrum of 2 × 2 matrices Linearization
We remark that the characteristic map µ: H → H, µ(λ) = c − (d − λ)b−1(a − λ) can be extended to a continuous map
µ: S 4 → S 4
on the sphere S 4 = H ∪ {∞}, because lim |µ(λ)| = ∞. |λ|→∞
Proposition
A polynomial map like µ and the power map λ2 are homotopic, hence they have the same topological degree, which equals 2.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 26 / 69 Spectrum of 2 × 2 matrices Linearization
Proposition For the characteristic map µ, we have
−1 −1 µ∗λ(X ) = Xb (a − λ) + (d − λ)b X . We shall classify the different possible spectra of 2 × 2 quaternionic matrices depending on the rank of this map. So, if we call
P = (d − λ)b−1, Q = b−1(a − λ),
we have XQ + PX = 0, that is, the so-called Sylvester equation.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 27 / 69 Spectrum of 2 × 2 matrices Sylvester equation
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 28 / 69 Spectrum of 2 × 2 matrices Sylvester equation
Sylvester equation
Let P, Q, R ∈ H. We want to study the rank of the R-linear map Σ: H → H given by Σ(X ) = PX + XQ.
The equation Σ(X ) = R has been widely studied, sometimes under the name of Sylvester equation [Bull. Am. Math. Soc., 1944].
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 29 / 69 Spectrum of 2 × 2 matrices Sylvester equation
The following Proposition is a reformulation of the results by Janovsk´aand Opfer [Mitt. Math. Ges. Hamb., 2008] .
Proposition
1 The rank of Σ is even, namely 0, 2 or 4; 2 rank Σ < 4 if and only if P and −Q are similar quaternions, that is they have the same norm and the same real part; 3 rank Σ = 0 if and only if P is a real number and Q = −P.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 30 / 69 Spectrum of 2 × 2 matrices Sylvester equation
Proof.– Let
P = t + xi + yj + zk Q = s + ui + vj + wk.
The matrix associated to Σ with respect to the basis {1, i, j, k} is
t + s −x − u −y − v −z − w x + u t + s −z + w y − v J = y + v z − w t + s −x + u z + w −y + v x − u t + s
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 31 / 69 Spectrum of 2 × 2 matrices Sylvester equation
Since we have an explicit expression for det J,
det J = (t + s)4 + 2(t + s)2(x 2 + y 2 + z2 + u2 + v 2 + w 2)+ (x 2 + y 2 + z2 − u2 − v 2 − w 2)2 ≥ 0,
it is easy to study the rank J.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 32 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 33 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
Classification of left spectra
Now we are in a position to reformulate the following result from Huang and So. a b Let λ ∈ be an eigenvalue of A = with b 6= 0. And, let us denote H c d
−1 −1 2 a0 = −b c, a1 = b (a − d), ∆ = a1 − 4a0.
Theorem The matrix A has one, two or infinite left eigenvalues.
The latter case is equivalent to the following conditions: a0, a1 are real numbers such that a0 6= 0 and ∆ < 0.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 34 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
Remark Let us call spherical the infinite case, because the spectrum
2 σl (A) = {(1/2)(a + d + bq): q = ∆}
2 is diffeomorphic to the sphere S ⊂ H0 = hi, j, ki.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 35 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
– Let λ ∈ H be an eigenvalue of A, that is µ(λ) = 0. In the next Propositions we shall apply the previous result about the Sylvester Equation to the differential Σ = µ∗λ given by
−1 −1 µ∗λ(X ) = Xb (a − λ) + (d − λ)b X .
Following the notation of Sylvester equation, let us remember that
P = (d − λ)b−1, Q = b−1(a − λ).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 36 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
Non-generic cases
Proposition (Rank 0)
If rank µ∗λ = 0, then a0, a1 are real numbers and ∆ = 0. Moreover λ equals (a + d)/2 and this is the only left eigenvalue of the matrix (this is a spherical degenerate spectrum).
Proof .– Since the rank is null, we know from the Sylvester Equation that
P = t ∈ R, Q = −t,
2 then a1 = −2t and 2λ = a + d. From µ(λ) = 0 it follows that a0 = +t , so ∆ = 0. Now it is easy to check that λ = a + tb is the only left eigenvalue of A.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 37 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
Proposition (Rank 2)
If rank µ∗λ = 2 two things may happen: 1 either the spectrum is spherical and all the eigenvalues of A are of rank 2; 2 or the matrix has just one eigenvalue.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 38 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
0 Proof .– By using the diffeomorphism a + bσl (A ) = σl (A) we can substitute A by the so-called “companion matrix”
0 1 A0 = . −a0 −a1
Now, we apply our proposition about the Sylvester Equation. Since the rank is 2, we have that
P = t + α, Q = −t + β
with α, β ∈ H0 = hi, j, ki, |α| = |β|= 6 0. Then
a1 = −2t + β − α.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 39 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
The first possibility is that β = α, then a1 = −2t. It follows from µ(λ) = 0 that
2 2 a0 = t + |β| 6= 0, ∆ = −4|β|2 < 0.
Then we have the so-called spherical case. In particular
2λ = −a1 + q, q = −2β.
The other eigenvalues have the form
2 2 (−a1 + q)/2, q = −4|β| ,
then the differential of µ verifies P = t − q−1/2 and Q = −t − q/2, and so they have rank 2 too.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 40 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
The second possibility is that β 6= α. Then
a1 = −2t + β − α,
a0 = (t + α)(t − β)
and the following lemma shows that the only eigenvalue is
λ = t − β.
Lemma Let A, B be two similar quaternions that do not commute. Then the equation λ2 − (A + B)λ + AB = 0 has the unique solution λ = B.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 41 / 69 Since the differential has maximal rank at the eigenvalue λ, the matrix A cannot be in the previous cases, hence all its eigenvalues are of rank 4. Then by the inverse function theorem the fiber µ−1(0) is discrete (in fact compact) and its cardinal equals the degree of the map µ, which is 2 (µ is homotopic to λ2 on the S 4). *Notice that the Jacobian is nonnegative.
Spectrum of 2 × 2 matrices Classification of left spectra
Generic case
Proposition (Rank 4)
If rank µ∗λ = 4 then the matrix has two different eigenvalues.
Proof .–
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 42 / 69 Then by the inverse function theorem the fiber µ−1(0) is discrete (in fact compact) and its cardinal equals the degree of the map µ, which is 2 (µ is homotopic to λ2 on the S 4). *Notice that the Jacobian is nonnegative.
Spectrum of 2 × 2 matrices Classification of left spectra
Generic case
Proposition (Rank 4)
If rank µ∗λ = 4 then the matrix has two different eigenvalues.
Proof .– Since the differential has maximal rank at the eigenvalue λ, the matrix A cannot be in the previous cases, hence all its eigenvalues are of rank 4.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 42 / 69 Spectrum of 2 × 2 matrices Classification of left spectra
Generic case
Proposition (Rank 4)
If rank µ∗λ = 4 then the matrix has two different eigenvalues.
Proof .– Since the differential has maximal rank at the eigenvalue λ, the matrix A cannot be in the previous cases, hence all its eigenvalues are of rank 4. Then by the inverse function theorem the fiber µ−1(0) is discrete (in fact compact) and its cardinal equals the degree of the map µ, which is 2 (µ is homotopic to λ2 on the S 4). *Notice that the Jacobian is nonnegative.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 42 / 69 3 × 3 matrices
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 43 / 69 – We shall find a map µA such that |µA(λ)| = Sdet(A − λI).
– µA will be in most cases a rational function instead of a polynomial.
3 × 3 matrices
The only known results about the left spectrum of 3 × 3 matrices are due to W. So [Southeast Asian Bull. Math., 2005]. W. So did a case by case study, depending on some relationships among the entries of the matrix. In general, there is no known method for solving the resulting equations.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 44 / 69 3 × 3 matrices
The only known results about the left spectrum of 3 × 3 matrices are due to W. So [Southeast Asian Bull. Math., 2005]. W. So did a case by case study, depending on some relationships among the entries of the matrix. In general, there is no known method for solving the resulting equations.
– We shall find a map µA such that |µA(λ)| = Sdet(A − λI).
– µA will be in most cases a rational function instead of a polynomial.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 44 / 69 3 × 3 matrices
Characteristic map a b c Consider the quaternionic matrix A = f g h ∈ M3×3(H). p q r
Let Pαβ be the real matrix obtained by interchanging the rows α, β in the identity matrix. Left (right) multiplication by this matrix permutes rows (columns).
Remark
Let A ∈ Mn×n(H) and π any permutation of the indices 1,..., n sending i to 1 and j to n. Then π can be written as a composition of transpositions Pαβ and the −1 resulting matrix PAP has the initial entry aij in the place (1, n).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 45 / 69 3 × 3 matrices Polynomial case
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 46 / 69 3 × 3 matrices Polynomial case
Polynomial case First, suppose that the matrix has a zero in the entry c = 0. Then
a − λ b 0 Sdet(A − λI) = Sdet f g − λ h . p q r − λ
There are three possibilities:
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 47 / 69 2 if b = 0 but h 6= 0, we reduce to the 2 × 2 case and we obtain
µ(λ) = q − (r − λ)h−1(g − λ) (a − λ);
3 finally, b 6= 0. We do a zero in the first row:
0 b 0 Sdet(A − λI) = Sdet f − (g − λ)b−1(a − λ) g − λ h , p − qb−1(a − λ) q r − λ
then
µ(λ) = p − qb−1(a − λ) − (r − λ)h−1 f − (g − λ)b−1(a − λ) .
3 × 3 matrices Polynomial case
1 if b, h = 0, we have a triangular matrix, so we can take
µ(λ) = (r − λ)(g − λ)(a − λ);
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 48 / 69 3 finally, b 6= 0. We do a zero in the first row:
0 b 0 Sdet(A − λI) = Sdet f − (g − λ)b−1(a − λ) g − λ h , p − qb−1(a − λ) q r − λ
then
µ(λ) = p − qb−1(a − λ) − (r − λ)h−1 f − (g − λ)b−1(a − λ) .
3 × 3 matrices Polynomial case
1 if b, h = 0, we have a triangular matrix, so we can take
µ(λ) = (r − λ)(g − λ)(a − λ);
2 if b = 0 but h 6= 0, we reduce to the 2 × 2 case and we obtain
µ(λ) = q − (r − λ)h−1(g − λ) (a − λ);
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 48 / 69 3 × 3 matrices Polynomial case
1 if b, h = 0, we have a triangular matrix, so we can take
µ(λ) = (r − λ)(g − λ)(a − λ);
2 if b = 0 but h 6= 0, we reduce to the 2 × 2 case and we obtain
µ(λ) = q − (r − λ)h−1(g − λ) (a − λ);
3 finally, b 6= 0. We do a zero in the first row:
0 b 0 Sdet(A − λI) = Sdet f − (g − λ)b−1(a − λ) g − λ h , p − qb−1(a − λ) q r − λ
then
µ(λ) = p − qb−1(a − λ) − (r − λ)h−1 f − (g − λ)b−1(a − λ) .
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 48 / 69 Suppose that aij = 0, with i 6= j. Then there is a real invertible matrix P such that the transformation PAP−1 does not change the spectrum and it gives a matrix with a13 = 0.
3 × 3 matrices Polynomial case
Theorem
If the matrix A ∈ M3×3(H) has some zero entry outside the diagonal, then it admits a polynomial characteristic map.
Proof .–
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 49 / 69 3 × 3 matrices Polynomial case
Theorem
If the matrix A ∈ M3×3(H) has some zero entry outside the diagonal, then it admits a polynomial characteristic map.
Proof .–
Suppose that aij = 0, with i 6= j. Then there is a real invertible matrix P such that the transformation PAP−1 does not change the spectrum and it gives a matrix with a13 = 0.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 49 / 69 3 × 3 matrices Rational case
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 50 / 69 3 × 3 matrices Rational case
Rational case Using the properties of Sdet, we can compute the Study’s determinant of the matrix A by doing zeroes in the first row. Then
0 0 c Sdet(A) = Sdet f − hc−1a g − hc−1b h . p − rc−1a q − rc−1b r
Definition
−1 We shall call pole of the matrix A ∈ M3×3(H) the point πA = g − hc b.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 51 / 69 2 for λ 6= πA we define
−1 µ(λ) = (πA − λ) p − (r − λ)c (a − λ) − −1 −1 −1 q − (r − λ)c b (πA − λ) f − hc (a − λ) .
3 × 3 matrices Rational case
We obtain the following characteristic map for A. Proposition
Let A be a matrix of order 3 × 3 such that c 6= 0. A characteristic map can be defined as follows: −1 1 if πA = g − hc b is the pole of A, then
−1 −1 µ(πA) = q − (r − πA)c b f − hc (a − πA) ;
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 52 / 69 3 × 3 matrices Rational case
We obtain the following characteristic map for A. Proposition
Let A be a matrix of order 3 × 3 such that c 6= 0. A characteristic map can be defined as follows: −1 1 if πA = g − hc b is the pole of A, then
−1 −1 µ(πA) = q − (r − πA)c b f − hc (a − πA) ;
2 for λ 6= πA we define
−1 µ(λ) = (πA − λ) p − (r − λ)c (a − λ) − −1 −1 −1 q − (r − λ)c b (πA − λ) f − hc (a − λ) .
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 52 / 69 However, the following example shows that µ may not be continuous at the pole πA.
3 × 3 matrices Rational case
Continuity.– Up to now we have defined maps which verify that
|µA(λ)| = Sdet(A − λI) for all λ ∈ H. Since Sdet is a continuous map we have
lim |µ(λ)| = |µ(πA)|. λ→πA
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 53 / 69 3 × 3 matrices Rational case
Continuity.– Up to now we have defined maps which verify that
|µA(λ)| = Sdet(A − λI) for all λ ∈ H. Since Sdet is a continuous map we have
lim |µ(λ)| = |µ(πA)|. λ→πA However, the following example shows that µ may not be continuous at the pole πA.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 53 / 69 3 × 3 matrices Rational case
Example
The pole πA = −i is not a left eigenvalue:
0 i 1 A = 3i − k 0 1 . k −1 + j + k 0
We have
µ(πA) = (−1 + j + k + 1)(3i − k − i) = 1 − i + 2j − 2k.
Let us take limits following a restricted direction:
lim µ(−i + εq) = −q(j + k)q−1(2i − k); ε→0
they depend on q ∈ H, hence the limit lim µ(λ) does not exist. λ→πA
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 54 / 69 Whether it is always possible or not to find a polynomial, or at least a continuous characteristic map for a given matrix A it is open to discussion. What we can say is that there are matrices which can not be reduced to the polynomial case c = 0.
3 × 3 matrices Rational case
We can prove that Theorem
The characteristic map µA is continuous if and only if the pole πA is a left eigenvalue of A.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 55 / 69 What we can say is that there are matrices which can not be reduced to the polynomial case c = 0.
3 × 3 matrices Rational case
We can prove that Theorem
The characteristic map µA is continuous if and only if the pole πA is a left eigenvalue of A.
Whether it is always possible or not to find a polynomial, or at least a continuous characteristic map for a given matrix A it is open to discussion.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 55 / 69 3 × 3 matrices Rational case
We can prove that Theorem
The characteristic map µA is continuous if and only if the pole πA is a left eigenvalue of A.
Whether it is always possible or not to find a polynomial, or at least a continuous characteristic map for a given matrix A it is open to discussion. What we can say is that there are matrices which can not be reduced to the polynomial case c = 0.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 55 / 69 3 × 3 matrices Rational case
Example (The pole πA = 1 + j is an eigenvalue) 0 −j i A = −1 + j j k , p q r with p, q, r ∈ H arbitrary. In fact,
µ(πA) = (q − (r − 1 − j)k)(−1 + j + j(−1 − j)) = 0.
*In fact, by moving around the entries outside the diagonal one can obtain up to six different characteristic maps, each one with a different pole.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 56 / 69 3 × 3 matrices Rational case
Extension to the infinite point.– We can prove that it is possible to extend the rational map µ defined in the c 6= 0 4 case to S = H ∪ {∞} (may be with a discontinuity at the pole πA).
Proposition The characteristic maps µ for the 3 × 3matrices verify that lim |µ(λ)| = ∞. |λ|→∞
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 57 / 69 3 × 3 matrices Topological study of the 3 × 3 case
1 Left eigenvalues Notion Study’s determinant Characteristic map
2 Spectrum of 2 × 2 matrices Topological degree Linearization Sylvester equation Classification of left spectra
3 3 × 3 matrices Polynomial case Rational case Topological study of the 3 × 3 case
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 58 / 69 2 We have the explicit expression of µ∗ for the two cases (polynomial and rational).
3 For a given eigenvalue λ of a matrix, we are able to compute the rank of µ∗λ (different eigenvalues may have µ∗λ with different rank).
3 × 3 matrices Topological study of the 3 × 3 case
Topological study of the 3 × 3 case
1 We cannot establish a complete classification.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 59 / 69 3 For a given eigenvalue λ of a matrix, we are able to compute the rank of µ∗λ (different eigenvalues may have µ∗λ with different rank).
3 × 3 matrices Topological study of the 3 × 3 case
Topological study of the 3 × 3 case
1 We cannot establish a complete classification.
2 We have the explicit expression of µ∗ for the two cases (polynomial and rational).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 59 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Topological study of the 3 × 3 case
1 We cannot establish a complete classification.
2 We have the explicit expression of µ∗ for the two cases (polynomial and rational).
3 For a given eigenvalue λ of a matrix, we are able to compute the rank of µ∗λ (different eigenvalues may have µ∗λ with different rank).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 59 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Now, we have to deal with a local notion of degree. The main result we need is the following Theorem Let Ω be a bounded open set in Rn. Let µ: Ω → Rn be a map continuous on the closure Ω and differentiable on Ω. Suppose that 0 is a regular value of µ and that 0 ∈/ µ(∂Ω). Then X deg(µ, Ω, 0) = sgn[Jµ(λ)] λ∈µ−1(0)
where we denote by Jµ the Jacobian of µ.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 60 / 69 2 if b = 0, h 6= 0 then
−1 −1 −1 µ∗λ(X ) = Xh (g−λ)(a−λ)− q − (r − λ)h (g − λ) X +(r−λ)h X (a−λ);
3 otherwise,
−1 −1 −1 µ∗λ(X ) = qb − (r − λ)h (g − λ)b X +Xh −1 f − (g − λ)b−1(a − λ) − (r − λ)h−1Xb −1(a − λ).
3 × 3 matrices Topological study of the 3 × 3 case
Proposition Let λ be a left eigenvalue of a matrix with c = 0. Then the differential of the polynomial characteristic map µA is given by 1 if b, h = 0 then
µ∗λ(X ) = −X (g − λ)(a − λ) − (r − λ)(g − λ)X − (r − λ)X (a − λ);
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 61 / 69 3 otherwise,
−1 −1 −1 µ∗λ(X ) = qb − (r − λ)h (g − λ)b X +Xh −1 f − (g − λ)b−1(a − λ) − (r − λ)h−1Xb −1(a − λ).
3 × 3 matrices Topological study of the 3 × 3 case
Proposition Let λ be a left eigenvalue of a matrix with c = 0. Then the differential of the polynomial characteristic map µA is given by 1 if b, h = 0 then
µ∗λ(X ) = −X (g − λ)(a − λ) − (r − λ)(g − λ)X − (r − λ)X (a − λ);
2 if b = 0, h 6= 0 then
−1 −1 −1 µ∗λ(X ) = Xh (g−λ)(a−λ)− q − (r − λ)h (g − λ) X +(r−λ)h X (a−λ);
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 61 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Proposition Let λ be a left eigenvalue of a matrix with c = 0. Then the differential of the polynomial characteristic map µA is given by 1 if b, h = 0 then
µ∗λ(X ) = −X (g − λ)(a − λ) − (r − λ)(g − λ)X − (r − λ)X (a − λ);
2 if b = 0, h 6= 0 then
−1 −1 −1 µ∗λ(X ) = Xh (g−λ)(a−λ)− q − (r − λ)h (g − λ) X +(r−λ)h X (a−λ);
3 otherwise,
−1 −1 −1 µ∗λ(X ) = qb − (r − λ)h (g − λ)b X +Xh −1 f − (g − λ)b−1(a − λ) − (r − λ)h−1Xb −1(a − λ).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 61 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Example (Three different eigenvalues, all of them with maximal rank 4) Let k 0 0 A = 3i − j −i i . 1 − 2k j −j The characteristic map is
µ(λ) = (−1 − k + λi)λ(k − λ)
hence σl (A) = {k, 0, −i − j}. The differential of µ at each eigenvalue is:
µ∗k(X ) = (−1 − i + k)X ;
µ∗0(X ) = −(1 + k)X k;
µ∗(−i−j)(X ) = −X (−1 − i + j) + X (i + j + k).
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 62 / 69 Regarding the differential, for the null eigenvalue it has maximal range, and for λ = −i − j it is null.
µ∗0(X ) = (1 + k)X (i + j);
µ∗(−i−j)(X ) = 0.
3 × 3 matrices Topological study of the 3 × 3 case
Example (Different rank of µ∗λ for different eigenvalues) −i − j 0 0 A = k −i i 1 − i j −j
A has two eigenvalues. This time,
µ(λ) = (1 + k − λi)λ(i + j + λ)
so σl (A) = {0, −i − j}.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 63 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Example (Different rank of µ∗λ for different eigenvalues) −i − j 0 0 A = k −i i 1 − i j −j
A has two eigenvalues. This time,
µ(λ) = (1 + k − λi)λ(i + j + λ)
so σl (A) = {0, −i − j}. Regarding the differential, for the null eigenvalue it has maximal range, and for λ = −i − j it is null.
µ∗0(X ) = (1 + k)X (i + j);
µ∗(−i−j)(X ) = 0.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 63 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Example (Three different eigenvalues) −1 1 0 A = j 0 −1 has three different eigenvalues 1 − j −1 1
λ1 = 0, q √ √ λ2, λ3 = ± −1 + 2 1 + 2 + j
because the characteristic map is
µ(λ) = −λ(2 + j) + λ3.
For λ1 the differential is the right translation µ∗0(X ) = −X (2 + j); for λ2 and λ3 it has the associated matrix 5 0 −5 0 0 7 0 1 M = , 5 0 5 0 0 −1 0 0
which is of maximal rank too.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 64 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Example (with µ∗ of odd range) j 1 0 A = 2i −k 1 . 2 − i − 2j −1 − j + k −i − k The characteristic map is
µ(λ) = 2 − i − 2j + (1 + j − k)(j − λ) + (i + k − λ) (2i + (k + λ)(j − λ)) .
For the left eigenvalue λ = 0 the differential is
µ∗0(X ) = kX + X i + (i + k)X j,
whose real associated matrix 0 −2 0 0 0 0 −2 0 M = 0 0 0 0 2 0 −2 0
has rank 3.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 65 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Conclusion.–
We have proved that a generic 2 × 2 quaternionic matrix has two different left eigenvalues. The singular cases are the spherical one (infinitely many eigenvalues) or a spectrum with just one point.
We have a characteristic map for any 3 × 3 quaternionic matrix (and µ∗!). The 3 × 3 case is still lacking a complete classification.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 66 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Deimling, K. Nonlinear functional analysis. Springer-Verlag, 1985. Eilenberg, S.; Steenrod, N. Foundations of algebraic topology. Princeton Mathematical Series, No.15, Princeton: University Press, XIV, 1952. Huang, L.; So, W. On left eigenvalues of a quaternionic matrix. Linear Algebra Appl., 323, No.1-3:105–116, 2001. Huang, L.; So, W. Quadratic formulas for quaternions. Appl. Math. Lett., 15, No.5:533–540, 2002. Janovsk´a,D.; Opfer, G. Linear equations in quaternionic variables. Mitt. Math. Ges. Hamb., 27:223-234, 2008.
M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 67 / 69 3 × 3 matrices Topological study of the 3 × 3 case
Mac´ıas-Virg´os,E.; Pereira-S´aez,M.J. Left eigenvalues of 2 × 2 symplectic matrices. Electron. J. Linear Algebra, 18:274–280, 2009. So, W. Quaternionic left eigenvalue problem. Southeast Asian Bull. Math. 29 No. 3:555-565, 2005. Wood, R.M.W. Quaternionic eigenvalues. Bull. Lond. Math. Soc., 17:137–138, 1985. Zhang, F. Quaternions and matrices of quaternions. Linear Algebra Appl., 251:21–57, 1997. Zhang, F. Gerˇsgorin type theorems for quaternionic matrices. Linear Algebra Appl., 424, No. 1:139–153, 2007.
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M.J. Pereira-S´aez(UDC) Left Eigenvalues of Quaternionic Matrices Sevilla, September 9-14, 2012 69 / 69